Confinement-Induced Symmetry Breaking of Active Surfaces

Abstract

The actomyosin cortex, a thin layer of a cross-linked polymer network near the cell surface, generates active forces that are responsible for cell shape changes. Many developmental processes that involve such cell shape changes, most prominently embryonic cell division, are spatially confined by eggshells. To investigate the potential role of confinement in redirecting active stresses and enabling symmetry breaking phenomena during cell shape transformations, we study a hydrodynamic minimal model in which the cell cortex is represented as an active fluid surface that undergoes symmetric division in the absence of confinement. When enclosed by an ellipsoidal shell, a spontaneous symmetry-breaking transition emerges at a critical degree of confinement, where symmetrically dividing surfaces become unstable and polarized geometries appear. We show that this transition is controlled by the tightness of the confinement and analyze the solution space of stationary surfaces to identify the mechanisms underlying confinement-induced symmetry breaking.

Figures (10)

• Figure 1: Confined symmetric divisions. (a) Surface dynamics (Movie 1) illustrating tangential surface flows $\tilde{v}_u=\bar{v}_u\tau_{\eta}/R_0$ (red arrows) and the normalized stress-regulator concentration $\tilde{c}=c/c_0$ under weak confinement ($\epsilon=0.64$, where $\epsilon=V_{\mathrm{cell}}/V_{\mathrm{shell}}$) and weak contractility ($\mathrm{Pe}=1.8$). (b) Same confinement as in (a), but with increased activity ($\mathrm{Pe}=13.5$) (Movie 2). (c) Dynamics with strong confinement ($\epsilon=0.87$) at the same contractility as in (b) ($\mathrm{Pe}=13.5$) (Movie 3). The transparent gray surface illustrates the confining shell. Right panels show profiles of $\tilde{v}_u$ and $\tilde{c}$ of the final steady states. All other parameters are listed in Table \ref{['tab:para_surface']}.
• Figure 2: Dynamics of confinement-induced symmetry breaking. (a) Representative evolution of a symmetry-breaking event showing stress regulator concentration $\tilde{c}=c/c_0$ (color scale) and tangential surface flows $\tilde{v}_u$ (red arrows). (b) Weakly polarized steady state in which the contractile ring stabilizes at an intermediate position between the pole and the equatorial plane ($\mathrm{Pe}=3.83$, Movie 4). (c) Time evolution of the location of maximal stress regulator concentration $u_{\mathrm{max}}(t)$. Orange triangle depicts the time corresponding to snapshot in (b). (d) One cycle of a persistent oscillatory state of asymmetric contractile ring formation and slipping ($\mathrm{Pe}=4.5$, Movie 5). (e) Temporal evolution of stress regulator concentration $\tilde{c}$ at pole $u=0$ (top) and $u_{\mathrm{max}}$ (bottom) for dynamics shown in (d). (f) One damped oscillation cycle ($i-iii$) followed by the final steady state ($iv$). ($\mathrm{Pe}=6.3$, Movie 6). Black arrows indicate magnitude of the normal velocity $\tilde{v}_n$ at the pole ($u=0$). (g) Temporal evolution of $\tilde{c}$ (top) and the normal velocity $\tilde{v}_n$ (bottom) at $u=0$ associated (f). Data shown uses confinement magnitude $\epsilon=0.87$, all other parameters are listed in Table \ref{['tab:para_surface']}SItext.
• Figure 3: Phase diagram and stationary shape spaces of confined active surfaces.(a) Phase diagram for the degree of confinement $\epsilon=V_{\mathrm{cell}}/V_{\mathrm{shell}}$ and the Péclet number $\mathrm{Pe}=(\xi\Delta\mu)_0 H_+(c_0)R_0^2/(D\eta_{\mathrm{b}})$ quantifying the strength of activity. Symbols depict numerical simulations (representative shapes at the bottom). Background shades indicate surface geometry: symmetric weakly (red) and strongly (blue) ingressed, and polarized (orange). Parameters used in Fig. \ref{['fig:Shell_effect']} are indicated by $\times$. Turning points of solution branches formed by symmetric, weakly ingressed surfaces (red lines in panels b--d) are denoted by $*$. Numerically determined pitchfork bifurcation points ($+$) agree well with critical points predicted from semi-analytical arguments ($\circ$, see Eq. \ref{['eq:growthrate']} and Fig. \ref{['fig:Pe*_Tran']} in SItext). (b--d) Pole-to-pole distance $\tilde{H}=H/R_0$ along stationary solution branches for different confinement degrees (black arrows in a). Solid (dashed) lines indicate stable (unstable) solutions and color shading as in panel a. Transitions to polarized surfaces (red to orange) and to strongly ingressed symmetric surfaces (red/orange to blue) in the phase diagram are determined by bifurcations and turning points, respectively, along symmetric, weakly ingressed surface branches (red lines). Oscillations occur for Pe bound by dashed lines in (c) and (d). Polar surfaces generally exhibit shorter pole-to-pole distances and can therefore more easily accommodate tight confinement. (e--h) Solution branches of weakly (e) and strongly (f) ingressed symmetric surfaces connect to corresponding polar surfaces via subcritical bifurcations (e,f) that become supercritical (g,h) when tightening the confinement. $\Delta u = u_{\mathrm{max}} - \tfrac{1}{2}$ quantifies the asymmetry of the contractile ring position and scales as $|\mathrm{Pe} - \mathrm{Pe}^*|^{1/2}$ in the vicinity of the bifurcation point (see Fig. \ref{['fig:Bifur_varAc_Root']} in SItext).
• Figure S1: Parameterization of the closed axisymmetric surface $\mathbf{X}(u,\phi,t)$. The transparent gray prolate surface represents the geometric confinement. The coordinates $u=0$ and $u=1$ correspond to the two poles of the surface, respectively. Under the SLE parameterization, the metric factor $h(u,t)=h(t)$ is independent of $u$ and equal to the total arclength $L$.
• Figure S2: Confinement-induced deformation and break-up of the Gibbs loop in the stationary solution space. At weak confinement ($\epsilon = 0.38,\,0.45$), solution branches form loops that continuously connects solutions corresponding to partially ingressed and fully ingressed geometries (see gao2025 for corresponding plot in the absence of any confinement). In contrast, at strong confinement ($\epsilon = 0.64,\,0.77,\,0.95$), these branches become disconnected. Insets show a zoomed-in view of the loop structure for $\epsilon = 0.38$. All other parameters are listed in Table \ref{['tab:para_surface']}.